arXiv research

Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…

A method to describe Riemann surfaces using graph profiles is proposed.

Study Liouville action for harmonic maps between Riemann surfaces.

Every open Riemann surface can be triangulated with equilateral triangles.

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

Study curve shortening flow on Riemann surfaces with conic singularities.

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by S…

Criterion found for Teichmüller extremal maps on infinite Riemann surfaces.

New invariant for Riemann surfaces connects to moduli space classes.

New method to parametrize infinite Riemann surfaces with bounded triangulations.

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

The paper discovers new ways Riemann surfaces can degenerate.

New formula and algorithm for computing distances on complex Riemann surfaces.

Estimates curvature for holomorphic maps on Riemann surfaces.

Algorithm constructs algebraic curves from translation surfaces.

Proves open Riemann surfaces can be embedded into 4D space.

Extends harmonic maps compactification to punctured Riemann surfaces.

Exact diameter found for some Riemann surfaces.

Study the boundary of Riemann surfaces with abelian automorphisms.

Criterion for Lie algebroid connections on compact Riemann surfaces.

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition funct…

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…

New proof shows special surfaces have finite type.

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…

Study interior estimates for solutions of Poisson equation on Riemann surfaces.

Sharp estimate shown to be rigid on curved surfaces.

This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

This article is an attempt to generalize Riemann's bilinear relations on compact Riemann surface of genus at least 2, which may lead to new structures in the theory of hyperbolic Riemann surfaces. No significant result is obtained, the article serves to bring the readers' attention to the observation made by [Bol-1949]…

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…

Optimizes the first eigenvalues of Riemann surfaces for large genus.

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

The paper studies geodesics on an icosahedron's Riemann surface.

Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comp…

Constructs metrics on Riemann surfaces with singularities.

The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action f…

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of th…

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

Analytic curves linked to algebraic ones via Schottky groups.

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

Scattering theory for harmonic one-forms on Riemann surfaces.

The paper describes superconformal structures on super Riemann surfaces using fatgraphs.